论文标题
2D经典海森堡模型的渗透和矢量值GFF的出口集
Percolation for 2D classical Heisenberg model and exit sets of vector valued GFF
论文作者
论文摘要
我们在本文中的动机是双重的。首先，我们研究了一类探索集的几何形状，称为退出集，它们与2D矢量值GFF：$ ϕ：z^2 \ to r^n，n \ geq 1 $相关。我们证明，有些令人惊讶的是，这些集合是A.S.退化为$ n \ geq 2 $，而当$ n = 1 $时，它们被推测为宏观和分形。 当$ n \ geq 2 $时，此分析可以理解$ \ {\ | ϕ（x）\ |，x \ in z^2 \} $的级别集的渗透属性随机电导给$ z^2 $的随机导电给出的旋转$ o（n）$。使用$ n $ - 矢量值GFF的出口集，我们在极限$β\ to \ infty $中获得了这种随机疾病的局部和几何描述。这使我们能够重新审视Patrascioiu和Seiler（[PS92，PS93，PS02]）的一系列著名作品，该作品反对Polyakov的预测，即当$ n \ geq 2 $（[POL75]）时，旋转$ O（n+1）$模型在所有温度下都是巨大的。我们使他们的论点的一部分是严格的，更重要的是，我们提供以下反例：我们建立了（任意）高电导率的千古环境，该环境（任意）小和断开的低电导率区域，尽管高电导率占优势，但$ XY $ $模型仍然很大。 我们非常关注，我们证明，在高$β$的情况下，经典海森堡模型的横向波动由$ n = 2 $ vectorial GFF给出。这在[POL75]中是隐含的，但我们在这里给出了第一个（非平凡的）严格证明。同样，与最近的工作[DF22]无关，我们表明旋转$ O（n）$模型的两点相关功能是根据任何$ n \ geq 1 $的电缆图中的某些渗透事件给出的。
Our motivation in this paper is twofold. First, we study the geometry of a class of exploration sets, called exit sets, which are naturally associated with a 2D vector-valued GFF : $ϕ: Z^2 \to R^N, N\geq 1$. We prove that, somewhat surprisingly, these sets are a.s. degenerate as long as $N\geq 2$, while they are conjectured to be macroscopic and fractal when $N=1$. This analysis allows us, when $N\geq 2$, to understand the percolation properties of the level sets of $\{\|ϕ(x)\|, x\in Z^2\}$ and leads us to our second main motivation in this work: if one projects a spin $O(N+1)$ model (classical Heisenberg model is $N=2$) down to a spin $O(N)$ model, we end up with a spin $O(N)$ in a quenched disorder given by random conductances on $Z^2$. Using the exit sets of the $N$-vector-valued GFF, we obtain a local and geometric description of this random disorder in the limit $β\to \infty$. This allows us to revisit a series of celebrated works by Patrascioiu and Seiler ([PS92, PS93, PS02]) which argued against Polyakov's prediction that spin $O(N+1)$ model is massive at all temperatures when $N\geq 2$ ([Pol75]). We make part of their arguments rigorous and more importantly we provide the following counter-example: we build ergodic environments of (arbitrary) high conductances with (arbitrary) small and disconnected regions of low conductances in which, despite the predominance of high conductances, the $XY$ model remains massive. Of independent interest, we prove that at high $β$, the transverse fluctuations of a classical Heisenberg model are given by a $N=2$ vectorial GFF. This is implicit in [Pol75] but we give here the first (non-trivial) rigorous proof. Also, independently of the recent work [DF22], we show that two-point correlation functions of the spin $O(N)$ model are given in terms of certain percolation events in the cable graph for any $N\geq 1$.
